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An All-Collapse Interpretation of Quantum Mechanics? |
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Not a soul but felt a fever of the mad and play'd |
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Some tricks of desperation. All but mariners |
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Plunged in the foaming brine and quit the vessel. |
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Shakespeare |
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According to the most common interpretation, quantum mechanics consists of two parts: (1) unitary evolution of the wave function for any isolated system in accord with the Schrodinger equation with some suitable Hamiltonian and initial conditions, and (2) the projection postulate along with the Born rule for probabilities, describing the outcome of a measurement or observation. This formulation has been remarkably successful, in the sense that it serves as a useful recipe for computing predictions, but it poses some interpretational difficulties. In practice we seem able to identify isolated quantum systems that evolve according to Schrodinger’s equation, and to distinguish those from “measurements”, which is to say, interactions with an apparatus that is represented in classical terms, and that lead to changes in the state of the original system according to the projection postulate, i.e., the so-called “collapse of the wave-function”. It is sometimes said that quantum mechanics consists of a theory describing how quantum systems interact with classical systems. However, a coherent realistic model for the projection (collapse) process has been elusive, and moreover the necessity of stipulating a classical system to establish the “basis” of the putative projection seems unsatisfactory, since it evidently precludes a purely quantum mechanical treatment of all systems. (The latter problem is obviously particularly acute when trying to apply quantum mechanics to cosmology.) |
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The combination of unitary evolution in accord with the Schrodinger equation and the collapse of the wave function in accord with the projection postulate could be seen as somewhat analogous to the two sides of the field equations of general relativity. The “left side” of the Einstein field equation is seen as pristine and definitive, whereas the “right side”, i.e., the stress-energy tensor, is seen as merely an ad hoc phenomenological provision, which is to say (in Einstein’s words) “a formal condensation of all things whose comprehension in the sense of a field theory is still problematic… a makeshift in order to give the general principle of relativity a preliminary closed-form expression”. In the view of many people, the Schrodinger equation is “the left side” of quantum mechanics, pristine and definitive, whereas the projection postulate and the Born rule for the probabilistic reduction of the wave function is “the right side”, a provisional makeshift to give the theory a preliminary closed-form expression. Accordingly there have been numerous ingenious attempts to rationalize or even eliminate the projection postulate from quantum mechanics (leading to so-called “no-collapse interpretations”), but none of these seems entirely satisfactory. |
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An alternative approach – one that has apparently received no attention – is to regard the projection process (or a suitable generalization of it) as fundamental, and then explain why this leads to the appearance of Schrodinger evolution in situations other than “measurements”. At first this might seem hopeless, because the projection process, as usually conceived, eliminates any superpositions, yielding a single definite value for the measured observable, and placing the observed system in a single definite eigenstate. This seems fundamentally inconsistent with the unitary evolution of the system into a superposition of observable states as given by the Schrodinger equation. However, it’s interesting to consider the possibility that the usual conception of the projection process may be just a particularly restrictive special case (which we call a measurement) of a more general process that actually does preserve some degree of superpositional information, depending on the circumstances, and that for an isolated (or nearly isolated) system this process approximates the state behavior usually associated with unitary evolution under the Schrodinger equation. |
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We’ve discussed elsewhere a generalization of linear system representations that emphasizes the underlying symmetry between the seemingly very distinct concepts of eigenvalues and eigenvectors. The projection process for a measurement in quantum mechanics corresponds to a highly specialized system of equations that is essentially designed to break the symmetry between eigenvalues and eigenvectors, so as to yield a single real-valued outcome within the measuring apparatus and a unique state vector for the measured system. Such an arrangement gives the (misleading) impression of a fundamental asymmetry between the measuring apparatus and the system being measured. When viewed in the context of generalized eigen systems it is seen to be just a specialized case of a fundamentally symmetrical interaction. The under-determination of such a system (in general) results in a multiplicity of possible outcomes that might be taken to represent a superposition of states. It would be interesting to know if, for a completely isolated system, this generalized projection process could yield behavior that approximates unitary evolution of the system’s state vector. |
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